A Tesla coil schematic is in the menu of Falstad's simulator. It has just one capacitor.
Although we would expect theoretically a single capacitor is needed, a second cap could be for purposes such as:
* some aspect of power factor correction?
* capacitors in series create reduced Farad value as the designer experimented with circuit operation
* a capacitor in each wire adds slightly greater isolation between transformers carrying high voltage, or carrying very different frequencies.
* to reduce possibility windings come into contact with output stage at volt levels that could lead to arcing.